# Four-dimensional vector multiplets in arbitrary signature

## TL;DR

This paper classifies four-dimensional ${
m N}=2$ supersymmetry algebras across various signatures, derives conditions for their isomorphism, and constructs vector multiplet representations revealing signature-dependent Lagrangian differences.

## Contribution

It provides a complete classification of four-dimensional ${
m N}=2$ supersymmetry algebras in arbitrary signatures and constructs associated vector multiplet representations.

## Key findings

- Unique supersymmetry algebras in Euclidean and neutral signatures.
- Two distinct algebras with different R-symmetry groups in Lorentz signature.
- Lagrangian sign differences linked to algebra isomorphisms and signatures.

## Abstract

We derive a necessary and sufficient condition for Poincar\'e Lie superalgebras in any dimension and signature to be isomorphic. This reduces the classification problem, up to certain discrete operations, to classifying the orbits of the Schur group on the vector space of superbrackets. We then classify four-dimensional ${\cal N}=2$ supersymmetry algebras, which are found to be unique in Euclidean and in neutral signature, while in Lorentz signature there exist two algebras with R-symmetry groups $\mathrm{U}(2)$ and $\mathrm{U(}1,1)$, respectively. By dimensional reduction we construct two off shell vector multiplet representations for each possible signature, and find that the corresponding Lagrangians always have a different relative sign between the scalar and the Maxwell term. In Lorentzian signature this is related to the existence of two non-isomorphic algebras, while in Euclidean and neutral signature the two theories are related by a local field redefinition which implements an isomorphism between the underlying supersymmetry algebras.